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Nonsingularity of matrices associated with classes of arithmetical functions
Journal of Algebra 281 (2004) 1–14 www.elsevier.com/locate/jalgebra
Nonsingularity of matrices associated with classes of arithmetical functions ✩ Shaofang Hong Mathematical College, Sichuan University, Chengdu 610064, PR China Department of Mathematics, Technion—IIT, Haifa 32000, Israel Received 10 April 2001
Communicated by Michel Broué
Abstract Let S = {x1 , . . . , xn } be a set of n distinct positive integers. Let f be an arithmetical function. Let [f (xi , xj )] denote the n × n matrix having f evaluated at the greatest common divisor (xi , xj ) of xi and xj as its i, j -entry and (f [xi , xj ]) denote the n × n matrix having f evaluated at the least common multiple [xi , xj ] of xi and xj as its i, j -entry. The set S is said to be gcd-closed if (xi , xj ) ∈ S for all 1 i, j n. For an integer x, let ν(x) denote the number of distinct prime factors of x. In this paper, by using the concept of greatest-type divisor introduced by S. Hong in [Adv. Math. (China) 25 (1996) 566–568; J. Algebra 218 (1999) 216–228], we obtain a new reduced formula for det f [(xi , xj )] if S is gcd-closed. Then we show that if S = {x1 , . . . , xn } is a gcd-closed set satisfying maxx∈S {ν(x)} 2, and if f is a strictly increasing (respectively decreasing) completely multiplicative function, or if f is a strictly decreasing (respectively increasing) completely multiplicative function satisfying 0 < f (p) 1/p (respectively f (p) p) for any prime p, then the matrix [f (xi , xj )] (respectively (f [xi , xj ])) defined on S is nonsingular. As a corollary, we show the following interesting result: The LCM matrix ([xi , xj ]) defined on a gcd-closed set is nonsingular if maxx∈S {ν(x)} 2. 2004 Elsevier Inc. All rights reserved. Keywords: Gcd-closed set; Greatest-type divisor; Arithmetical function; Nonsingularity
✩ Supported partially by an NNSF of China (Grant No. 10101015) and the Lady Davis Fellowship at the Technion. E-mail addresses: [email protected], [email protected].
0021-8693/$ – see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jalgebra.2004.07.026
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S. Hong / Journal of Algebra 281 (2004) 1–14
1. Introduction Let S = {x1 , . . . , xn } be a set of n distinct positive integers. The matrix having the greatest common divisor (xi , xj ) of xi and xj as its i, j -entry is called the greatest common divisor (GCD) matrix, denoted by [(xi , xj )]. The matrix having the least common multiple [xi , xj ] of xi and xj as its i, j -entry is called the least common multiple (LCM) matrix, denoted by ([xi , xj ]). The set is said to be factor-closed if it contains every divisor of x for any x ∈ S. H.J.S. Smith [18] showed that the determinant of the GCD matrix [(xi , xj )] on a factor-closed set S is the product ni=1 ϕ(xi ), where ϕ is Euler’s totient function. The set S is said to be gcd-closed if (xi , xj ) ∈ S for all 1 i, j n. It is clear that a factor-closed set is a gcd-closed set but not conversely. In [3], Beslin and Ligh generalized Smith’s result by showing that the determinant of GCD matrix [(xi , xj )] defined on a gcd-closed set S = {x1 , . . . , xn } is equal to the product nk=1 αk , where αk =
ϕ(d).
d|xk dxt , xt
In [4], Bourque and Ligh proved that the determinant of the LCM matrix ([xi , xj ]) defined on a gcd-closed set S = {x1 , . . . , xn } is equal to the product nk=1 xk2 βk , where βk =
g(d)
d|xk dxt , xt
with the arithmetical function g defined by g(m) = m1 d|m dµ(d), and the function µ is the Möbius function. Let f be an arithmetical function. Let [f (xi , xj )] denote the n×n matrix having f evaluated at the greatest common divisor (xi , xj ) of xi and xj as its i, j -entry and (f [xi , xj ]) denote the n × n matrix having f evaluated at the least common multiple [xi , xj ] of xi and xj as its i, j -entry. In [18], Smith also considered the determinant of the matrix [f (xi , xj )] on a factor-closed set S. It was shown to be the product nk=1 (f ∗ µ)(xk ), where f ∗ µ is the Dirichlet product of f and µ. Apostol [1] extended Smith’s result. McCarthy in [17] generalized Smith’s and Apostol’s results to the class of even functions (mod r). Then Bourque and Ligh [6] extended Smith’s and Apostol’s and McCarthy’s results to certain two-variable arithmetical function. Hong [12] improved the lower bounds for the determinants of matrices introduced by Bourque and Ligh. Hong [13] extended the results of Smith, of Apostol, of McCarthy, and of Bourque and Ligh. Bourque and Ligh [4] showed that the GCD matrix [(xi , xj )] on S divides the LCM matrix ([xi , xj ]) on S in the ring Mn (Z) of n × n matrices over the integers if S is factorclosed. Hong [14] proved that such a factorization is no longer true in general if S is gcdclosed. In fact, Hong showed that if n 3, then for any gcd-closed set S = {x1 , . . . , xn }, the GCD matrix on S divides the LCM matrix on S in the ring Mn (Z). For n 4, there exists a gcd-closed set S = {x1 , . . . , xn }, such that the GCD matrix on S does not divide the LCM matrix on S in the ring Mn (Z). From Bourque and Ligh’s result [7, Theorem 4],
S. Hong / Journal of Algebra 281 (2004) 1–14
3
we can see that if S is a factor-closed set and f is a multiplicative function such that (f ∗ µ)(d) ∈ Z∗ whenever d | lcm(S) where Z∗ := Z\{0} denotes the set of nonzero integers and lcm(S) means the least common multiple of all elements in S, then the matrix (f (xi , xj )) divides the matrix (f [xi , xj ]) in the ring Mn (Z). Recently, Hong [15] show that for any multiple-closed set S (namely, y ∈ S whenever x | y | lcm(S) for any x ∈ S), and for any divisor chain S (i.e. x1 | · · · | xn ), if f is a completely multiplicative function such that (f ∗ µ)(d) is a nonzero integer whenever d | lcm(S), then the matrix (f (xi , xj )) divides the matrix (f [xi , xj ]) in the ring Mn (Z). But such a factorization is no longer true if f is multiplicative. A general question, raised in a slightly different form by Bourque and Ligh at the end of [5] is to characterize the couples (S, f ) with S gcd-closed for which the matrix [f (xi , xj )] is nonsingular. They gave tool to attack this problem by giving the following formula, which is another generalization of Smith’s result. Theorem 1.1 [5]. The determinant of the matrix [f (xi , xj )] defined on a gcd-closed set S = {x1 , . . . , xn } is equal to x∈S αS,f (x), where
αS,f (x) :=
(f ∗ µ)(d) and
(1)
d|x d∈ / ES (x)
ES (x) := z ∈ Z+ : ∃y ∈ S, y < x, z | y with Z+ denoting the set of positive integers. Thus the problem is reduced to recognize if x∈S αS,f (x) = 0 or not. Let A be a set of positive integers. We denote by gcd(A) the greatest common divisor of the elements of A. Moreover, for x ∈ A, we say that d ∈ A is a greatest-type divisor of x in A, if d | x, d = x and the conditions d | y | x and y ∈ A imply that y ∈ {x, d} (see [11]). Note that the concept of greatest-type divisor introduced in [11] played a central role in our solution [11] to the Borque–Ligh conjecture [4]. We define GA (x) to be the set of all greatest-type divisors of x in A. Namely, we have GA (x) := d ∈ A: d | x, d = x and [d | y | x and y ∈ A] ⇒ y ∈ {x, d} . The aim of our paper is to continue the work of Bourque and Ligh [5] by giving the following formula. Theorem 1.2. For all gcd-closed sets S and all complex-valued functions f defined on S and x ∈ S, we have αS,f (x) =
(−1)|J | f gcd(J ∪ {x}) .
J ⊂GS (x)
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S. Hong / Journal of Algebra 281 (2004) 1–14
(Notice that if J = ∅, gcd(J ∪ {x}) = gcd(J ).) This formula makes the computation of αS,f (x) easy if |GS (x)| is small. For instance in the case where S is a divisor chain or when all the x are primes or one, because then |GS (x)| = 1. For an integer x, let ν(x) denote the number of distinct prime factors of x. If maxx∈S {ν(x)} 2, then GS (x) even if it is big, has a special structure that permits to give sufficient condition on f such that the matrix [f (xi , xj )] is nonsingular. An arithmetical function f (x) is said to be strictly increasing (respectively strictly decreasing) if we have f (x1 ) < f (x2 ) (respectively f (x1 ) > f (x2 )) whenever x1 < x2 and x1 , x2 ∈ Z+ . Theorem 1.3. Let S = {x1 , . . . , xn } be a gcd-closed set satisfying maxx∈S {ν(x)} 2 and f a completely multiplicative function. If f is a strictly increasing function or f is a strictly decreasing function satisfying 0 < f (p) 1/p for all primes p, then the matrix [f (xi , xj )] on S is nonsingular. As we shall see, it is easy to deduce the following result. Theorem 1.4. Let S = {x1 , . . . , xn } be a gcd-closed set satisfying maxx∈S {ν(x)} 2 and f a completely multiplicative function. If f is a strictly increasing function satisfying f (p) p for all primes p or f is a strictly decreasing function, then the matrix (f [xi , xj ]) on S is nonsingular. As a consequence, we have the following result. Theorem 1.5. Let S = {x1 , . . . , xn } be a gcd-closed set satisfying maxx∈S {ν(x)} 2 and let ε < 0 or ε 1. Then the matrix ([xi , xj ]ε ) on S is nonsingular. In particular, we have the following interesting result. Theorem 1.6. Let S = {x1 , . . . , xn } be a gcd-closed set satisfying maxx∈S {ν(x)} 2. Then the LCM matrix ([xi , xj ]) on S is nonsingular. Moreover, we will show by an example (see Section 4) that we cannot change 2 by 3 in this last theorem. Theorem 1.2 is proved in the second section and the proofs of Theorems 1.3–1.5 are given in the third section. The last section is devoted to some remarks and conjectures. Throughout this paper, we denote by |A| the cardinality of a finite set A.
2. Proof of Theorem 1.2 We [11] gave a reduction of the formula for the determinant of the LCM matrix ([xi , xj ]) by introducing the concept of the greatest-type divisor. In the present section we will prove Theorem 1.2 by using the similar ideas as in [8,11]. First one needs a generalization of the principle of cross-classification in [8] to give a preliminary reduction of the formula
S. Hong / Journal of Algebra 281 (2004) 1–14
5
for αS,f . Note that an alternative proof using induction was given in [14]. For the convenience of the reader, we here give another proof using the principal of cross-classification. In next lemma, for a subset T of R, we denote by T the set R \ T of the elements of R that are not in T . Moreover, j ∈φ Rj is meant to be R. Lemma 2.1 [8, Lemma 1]. Let R be any given finite set, {Ri }i∈I a finite family of subsets of R, and f a complex-valued function defined on R. Then f (x) = (−1)|J | f (x). (2) x∈
i∈I
J ⊂I
Ri
x∈
j∈J
Rj
Proof. First observe that for f = 1, (2) is the principal of cross-classification [2] that can be written as |J | Ri = (−1) Rj . J ⊂I
i∈I
j ∈J
Let A be a subset of R. Define the function 1A for x ∈ R by 1A (x) = 1 if x ∈ A, and 0 otherwise. By applying the principal of cross-classification to the family of |I | + 1 subsets, {Ri }i∈I and A, we get
|J | |J |+1 1A (x) = R i ∩ A = (−1) Rj + (−1) Rj ∩ A x∈
i∈I
Ri
J ⊂I
i∈I
j ∈J
J ⊂I
j ∈J
= (−1)|J | Rj − Rj ∩ A = (−1)|J | Rj ∩ A J ⊂I
=
J ⊂I
j ∈J
(−1)
j ∈J
|J | x∈
j∈J
J ⊂I
j ∈J
1A (x). Rj
This completes the proof of (2) for f = 1A . Now the formula f= f (y)1{x} y∈R
permits us to deduce (2) for general f , by linearity. 2 We can now give a preliminary reduction of the formula for αS,f (x). Claim 2.2. Let S = {x1 , . . . , xn } be gcd-closed and x ∈ S. Then (−1)|J | f gcd(J ∪ {x}) , αS,f (x) = J ⊂TS (x)
where TS (x) := {y: y ∈ S and y < x}.
(3)
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S. Hong / Journal of Algebra 281 (2004) 1–14
Proof. In Lemma 2.1, let R = {d ∈ Z+ : d | x}. For y ∈ TS (x), let RS (y) = {d ∈ R: d | y}. Then RS (y) = {d ∈ Z+ : d | (x, y)} and y∈TS (x) RS (y) = ES (x) ∩ R. By Lemma 2.1, we have (f ∗ µ)(d) = (−1)|J | (f ∗ µ)(d). (4) αS,f (x) = J ⊂TS (x)
d∈ES (x)∩R
Since d ∈
y∈J
d∈
y∈J
RS (y)
RS (y) implies that d | gcd(J ∪ {x}), we have by [10, Lemma 1]: d∈
y∈J
(f ∗ µ)(d) = f gcd(J ∪ {x}) .
(5)
RS (y)
It then follows from Eqs. (4) and (5) that (3) holds. This completes the proof of Claim 2.2. 2 Consequently, we give further reduction of the formula for αS,f (x). Claim 2.3. Let S = {x1 , . . . , xn } be gcd-closed and x ∈ S. Then
αS,f (x) =
(−1)|J | f gcd(J ∪ {x}) ,
(6)
J ⊂JS (x)
where JS (x) := TS (x) \ IS (x) and IS (x) := {y ∈ TS (x): y x}. Proof. If |IS (x)| = 0, then it follows from Claim 2.2 that Claim 2.3 holds. In what follows let |IS (x)| 1. Note that for y ∈ JS (x), we have y | x. Since S is gcd-closed, min(S) | x. Thus one has |JS (x)| 1. Note also that |IS (x)| + |JS (x)| = |TS (x)|. By Claim 2.2, we have αS,f (x) = f (x) + ∆ + ∆, where ∆ = (−1)|J | f gcd(J ∪ {x}) and
∆=
J ⊂JS (x) |J |1
(−1)|J1 |+|J2 | f gcd(J1 ∪ J2 ∪ {x}) .
(7)
(8)
J1 ⊂JS (x) J2 ⊂IS (x) |J1 |1 |J2 |1
For any given subset J2 ⊂ IS (x) with |J2 | 1, it follows from S is gcd-closed that gcd(J2 ∪ {x}) ∈ S. Let l = gcd(J2 ∪ {x}). Then l | x and l < x. So l ∈ JS (x). Thus we have by (8): ∆=
J2 ⊂IS (x) J1 ⊂JS (x) |J2 |1 |J1 |1
(−1)|J1 |+|J2 | f gcd(J1 ∪ J2 ∪ {x})
S. Hong / Journal of Algebra 281 (2004) 1–14
=
(−1)|J1 |+|J2 | f gcd(J1 ∪ J2 ∪ {x})
J2 ⊂IS (x) J1 ⊂JS (x) l ∈J / 1 |J2 |1
=
7
+ (−1)|J1 |+|J2 |+1 f gcd(J1 ∪ J2 ∪ {l} ∪ {x}) (−1)|J1 |+|J2 | f gcd(J1 ∪ {l}) + (−1)|J1 |+|J2 |+1 f gcd(J1 ∪ {l})
J2 ⊂IS (x) J1 ⊂JS (x) l ∈J / 1 |J2 |1
= 0.
(9)
Therefore, it follows from Eqs. (7) and (9) that (6) holds. The proof of Claim 2.3 is complete. 2 Now we can prove Theorem 1.2. Proof of Theorem 1.2. For the case |TS (x)| 1, Theorem 1.2 is clearly true. In what follows, let |TS (x)| 2. Let JS (x) = {y ∈ TS (x): y | x}. Then |JS (x)| 1. It is clear that GS (x) ⊂ JS (x). If |JS (x)| = 1, then JS (x) = {min(S)}. Note that |GS (x)| 1. So one has GS (x) = {min(S)} = JS (x). Thus by Claim 2.3, the result is true. In the following let |JS (x)| 2. Let LS (x) = JS (x)\GS (x). We assert that LS (x) = ∅. Otherwise, LS (x) = φ implies that GS (x) = JS (x). But min(S) ∈ JS (x). Then min(S) ∈ GS (x). From |JS (x)| 2, one deduces that there is y ∈ JS (x), such that y ∈ JS (x) = GS (x). Since S is gcdclosed, we have min(S) | y. This is absurd. Therefore, the assertion is true. In a similar way to (7), we have by Claim 2.3: αS,f (x) = f (x) + ∆ + ∆, where ∆ = (−1)|J | f gcd(J ∪ {x}) and
∆=
J ⊂GS (x) |J |1
J1 ⊂GS (x) J2 ⊂LS (x) |J2 |1
=
J2 ⊂LS (x) |J2 |1
(−1)|J2 |
(−1)|J1 |+|J2 | f gcd(J1 ∪ J2 ∪ {x})
(−1)|J1 | f gcd(J1 ∪ J2 ∪ {x}) .
(10)
J1 ⊂GS (x)
To prove Theorem 1.2, one needs only to show that ∆ = 0 which will be done in the following. For any given J2 ⊂ LS (x) with |J2 | 1, let P(x, J2) = {y ∈ GS (x): z | y for some z ∈ J2 } and let Q(x, J2 ) = GS (x) \ P(x, J2). Clearly we have 1 |P(x, J2 )| |GS (x)| and 0 |Q(x, J2 )| |GS (x)| − 1. Then
(−1)|J1 | f gcd(J1 ∪ J2 ∪ {x})
J1 ⊂GS (x)
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S. Hong / Journal of Algebra 281 (2004) 1–14
=
(−1)|J1 |+|J1 | f gcd J1 ∪ J1 ∪ J2 ∪ {x}
J1 ⊂Q(x,J2 ) J1 ⊂P (x,J2 )
=
(−1)|J1 |+|J1 | f gcd J1 ∪ J2 ∪ {x}
J1 ⊂Q(x,J2 ) J1 ⊂P (x,J2 )
(since by the definition of P(x, J2 ), we have gcd(J1 ∪ J2 ) = gcd(J2 ) for any J1 ⊂ P(x, J2 ))
=
(−1)|J1 | f gcd J1 ∪ J2 ∪ {x}
J1 ⊂Q(x,J2 )
=
=
J1 ⊂Q(x,J2 )
=
(−1)|J1 |
J1 ⊂P (x,J2 )
(−1)
|J1 |
|P (x,J2 )| f gcd J1 ∪ J2 ∪ {x} 1 + (−1)r
J1 ⊂Q(x,J2 )
r=1
(−1)
|J1 |
1
J1 ⊂P (x,J2 ) |J1 |=r
|P (x,J2 )| r |P(x, J2 )| f gcd J1 ∪ J2 ∪ {x} 1 + (−1) r r=1
(−1)|J1 | f gcd J1 ∪ J2 ∪ {x} · (1 − 1)|P (x,J2)|
J1 ⊂Q(x,J2 )
= 0.
(11)
It then follows from Eqs. (10) and (11) that ∆ = 0. This completes the proof of Theorem 1.2. 2
3. Proofs of Theorems 1.3–1.5 Throughout this section, let S = {x1 , . . . , xn } be a gcd-closed set. Let αS,f (x) be defined as in (1). It is clear that αS,f (min(S)) = f (min(S)). We have the following lemmas. Lemma 3.1. Let x = pe q h ∈ S, where p and q are distinct primes, e and h are positive integers. Then GS (x) must have the form {pe1 q h1 , . . . , pem q hm }, where 1 m min{e, h}, 0 e1 < · · · < em e, h h1 > · · · > hm 0, and ei + hi e + h − 1 (i = 1, . . . , m). Proof. Let |GS (x)| = m. Since x = pe q h , one may let GS (x) = {pe1 q h1 , . . . , pem q hm }, where ei and hi (1 i m) are nonnegative integers satisfying 0 ei e, 0 hi h and ei + hi e + h − 1. We claim that for any i, j ∈ {1, . . . , m}, i = j , we have ei = ej . Otherwise, suppose that there exist i, j ∈ {1, . . . , m}, i = j , such that ei = ej . Then pei q hi | pej q hj or pej q hj | pei q hi . This contradicts to the fact that pei q hi and pej q hj are the greatest-type divisors of x in S. Thus ei = ej for any i, j ∈ {1, . . . , m}, i = j . Similarly, for i, j ∈ {1, . . . , m}, i = j , we have hi = hj .
S. Hong / Journal of Algebra 281 (2004) 1–14
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Without loss of generality, one may assume that 0 e1 < e2 < · · · < em . Since pe1 q h1 , . . . , pem q hm are the greatest-type divisors, then for any i, j ∈ {1, . . . , m}, i = j , we have both pei q hi pej q hj and pej q hj pei q hi . Therefore for any i ∈ {1, 2, . . . , m − 1}, it follows from ei < ei+1 and pei q hi pei+1 q hi+1 that hi > hi+1 . So h1 > h2 > · · · > hm 0. It is clear that for 1 i m, we have ei + hi e + h. Suppose that there exists 1 i m such that ei + hi = e + h. One can deduce that ei = e and hi = h. So we have pei q hi = x. This contradicts that pei q hi is a greatest-type divisor of x. Then for i = 1, 2, . . . , m, we have ei + hi e + h − 1. The proof of Lemma 3.1 is complete. 2 Lemma 3.2. Let x = pe q h ∈ S, where p and q are distinct primes, e and h are positive integers. If GS (x) = {pe1 q h1 , . . . , pem q hm }, where 1 m min{e, h}, 0 e1 < · · · < em e, h h1 > · · · > hm 0, and ei + hi e + h − 1 (i = 1, . . . , m), then we have e h e h 1 1 f p q − f p q , m αS,f (x) = e h m−1 e h f p iq i + f pei q hi+1 , f p q − i=1
if m = 1, if m 2.
i=1
Proof. By Lemma 3.1, we know that GS (x) = pej q hj : 1 j m with 0 e1 < · · · < em e and h h1 > · · · > hm 0. We can write the parts J of GS (x) as J = pei q hi : i ∈ I
with I ⊂ {1, . . . , m}.
With the notation (emin(φ) , hmin(φ) ) = (e, h), Theorem 1.2 gives
αS,f (x) =
(−1)|I | f pemin(I ) q hmax(I )
I ⊂{1,...,m}
= f (x) +
f pea q hb
(−1)|I | .
min(I )=a max(I )=b
1abm
We conclude by the observation that for b a + 2, we have min(I )=a max(I )=b
(−1)|I | =
I ⊂{a+1,...,b−1}
This completes the proof of Lemma 3.2. 2 We are now in a position to prove Theorem 1.3.
(−1)|I | = 0.
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S. Hong / Journal of Algebra 281 (2004) 1–14
Proof of Theorem 1.3. Without loss of any generality, one may let x1 < · · · < xn . First we note that f (x) > 0 for any integer x > 0, since f is completely multiplicative satisfying f (p) > 0 for all prime p. Since maxx∈S {ν(x)} 2, then for x ∈ S with x > 1, we have ν(x) = 1 or 2. That is, we have x = pe , where e 1 is an integer and p is a prime, or x = pe q h , where e 1 and h 1 are integers, p and q are distinct primes. We claim that for x ∈ S, αS,f (x) = 0. If x = x1 , then by Theorem 1.2 we have αS,f (x) = f (x1 ) = 0. The claim is true. In the following, let x = x1 . Consider the following two cases. Case 1. x = pe . Then x has only one greatest-type divisor in S whose form must be pl , where l is an integer and 0 l e − 1. By Theorem 1.2, we have αS,f (x) = f (pe ) − f (pl ). Since f is strictly increasing or strictly decreasing, one then deduce that αS,f (x) > 0 or αS,f (x) < 0, respectively. The claim is true. Case 2. x = pe q h . It follows from Lemmas 3.1 and 3.2 that there exist 2m, where 1 m min{e, h}, integers e1 , . . . , em , h1 , . . . , hm satisfying 0 e1 < · · · < em e, h h1 > · · · > hm 0, and ei + hi e + h − 1 (i = 1, . . . , m), such that GS (x) = {pe1 q h1 , . . . , pem q hm }. If m = 1, then by Lemma 3.2 we have αS,f (x) = f (pe q h ) − f (pe1 q h1 ). By the assumption that f is strictly decreasing or strictly increasing, we have αS,f (x) < 0 or αS,f > 0, respectively. The claim is true. In the following let m 2. Let f be a strictly increasing function. Then for primes p and q, f (p) > f (1) = 1 and f (q) > 1. Since em e, h1 h, we have f (pe q h ) f (pem q h1 ). Therefore, m m−1 αS,f (x) f pem q h1 − f pei q hi + f pei q hi+1 i=1
=
m−1
i=1
e h f p m q i − f pem q hi+1 − f pei q hi + f pei q hi+1
i=1
=
m−1
f pei q hi+1 f (p)em −ei − 1 f (q)hi −hi+1 − 1 > 0.
i=1
So the claim for this case is true. Now let f be a strictly decreasing function satisfying 0 < f (p) 1/p for all primes p. If m = 2, then by Lemma 3.2 we have αS,f (x) = f pe q h + f pe1 q h2 − f pe1 q h1 − f pe2 q h2 . Since f is completely multiplicative, we have αS,f (x) = f pe q h + f pe2 q h1 f pe1 −e2 q h2 −h1 − f pe1 −e2 − f q h2 −h1 1 1 1 − − . = f pe q h + f pe2 q h1 f (pe2 −e1 q h1 −h2 ) f (pe2 −e1 ) f (q h1 −h2 )
S. Hong / Journal of Algebra 281 (2004) 1–14
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Since f satisfies 0 < f (p) 1/p for all primes p, one can then deduce that 1 f (pe2 −e1 q h1 −h2 )
−
1 f (pe2 −e1 )
−
1 f (q h1 −h2 )
=
1
−1
1
f (pe2 −e1 ) f (q h1 −h2 ) pe2 −e1 − 1 q h1 −h2 − 1 − 1
−1 −1
(2 − 1)(3 − 1) − 1 = 1 > 0. Thus αS,f (x) > 0. The claim holds for this case. In what follows let m 3. Then we have by Lemma 3.2: αS,f (x) = f pe q h + f pe1 q h2 − f pe1 q h1 − f pe2 q h2 +
m e f p i−1 q hi − f pei q hi . i=3
By the above arguments, one knows that f pe1 q h2 − f pe1 q h1 − f pe2 q h2 > 0. Since f is strictly decreasing, we have for 3 i m: f pei−1 q hi − f pei q hi > 0. So αS,f (x) > 0. The claim is true in this case. It then follows from the claim and Theorem 1.1 that det[f (xi , xj )] = 0. Therefore the matrix [f (xi , xj )] is nonsingular. The proof of Theorem 1.3 is complete. 2 Finally we give the proofs of Theorems 1.4 and 1.5. Proof of Theorem 1.4. Define for any integer x the function 1/f to be 0 if f (x) = 0; 1/f (x) if f (x) = 0. Clearly we have
1 (f [xi , xj ]) = diag{f (x1 ), . . . , f (xn )} · (xi , xj ) · diag{f (x1 ), . . . , f (xn )}. f Then Theorem 1.3 applied to the function 1/f gives the desired result.
2
Proof of Theorem 1.5. Let f (x) = x ε in Theorem 1.4, where ε < 0 or ε 1. Then the result follows immediately from Theorem 1.4. 2
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4. Remarks and conjectures In [4], Bourque and Ligh conjectured that the LCM matrix ([xi , xj ]) defined on any gcd-closed set S = {x1 , . . . , xn } is nonsingular. In [8,11], by introducing the concept of greatest-type divisor to reduce greatly the formula for det[xi , xj ], we showed that this conjecture is true for n 7 and is not true for n 8. We showed also, in [9], that this conjecture is true for certain gcd-closed sets. From Theorem 1.6, one then knows that the Bourque–Ligh conjecture is true for the gcd-closed sets S = {x1 , . . . , xn } so that each of S has at most two distinct prime factors. However, if S is a gcd-closed set satisfying maxx∈S {ν(x)} 3, then the LCM matrix [(xi , xj )] need not be nonsingular. For example, let S = {1, 2, 5, 8, 10, 13, 26, 65, 520}.
(12)
Then S is gcd-closed and maxx∈S {ν(x)} = 3. Since GS (520) = {8, 10, 26, 65}, we have β9 (1, 2, 5, 8, 10, 13, 26, 65, 520) =
1 1 1 1 1 1 1 − − − − + + = 0. 520 8 10 26 65 5 13
Thus the LCM matrix ([xi , xj ]) on S is singular. We believe that Theorems 1.3 and 1.4 still hold if the conditions “satisfying 0 < f (p) 1/p for all primes p” and “satisfying f (p) p for all primes p” are suppressed, respectively. We believe also that the condition “ε < 0 or ε 1” in Theorem 1.5 could be improved to “ε = 0.” Unfortunately, so far we have not found the proof yet. The set S is said to be odd-gcd-closed if S is gcd-closed and every element in S is an odd number. The set S is said to be even-gcd-closed if S is not odd-gcd-closed. By [11] and (12), we know that there are even-gcd-closed sets S such that the LCM matrix ([xi , xj ]) on S is singular. But it is not clear that there is an odd-gcd-closed set S such that the LCM matrix ([xi , xj ]) on S is singular. We believe that the answer to this question should be negative. We raise the following conjecture which generalizes Conjecture 5.1 of [16]. Conjecture 4.1. Let ε = 0 and let S = {x1 , . . . , xn } be an odd-gcd-closed set. Then the matrix ([xi , xj ]ε ) on S is nonsingular. An arithmetical function f is called strictly monotonous if it is strictly increasing or strictly decreasing. Furthermore, we propose the following conjectures. Conjecture 4.2. Let S = {x1 , . . . , xn } be an odd-gcd-closed set and f a completely multiplicative function. If f is a strictly monotonous function, then the matrix (f (xi , xj )) on S is nonsingular. Conjecture 4.3. Let S = {x1 , . . . , xn } be an odd-gcd-closed set and f a completely multiplicative function. If f is a strictly monotonous function, then the matrix (f [xi , xj ]) on S is nonsingular.
S. Hong / Journal of Algebra 281 (2004) 1–14
13
According to [14], there is a gcd-closed set S = {x1 , . . . , xn } with maxx∈S {|GS (x)|} = 2 such that the GCD matrix ((xi , xj )) on S does not divide the LCM matrix ([xi , xj ]) on S in the ring Mn (Z). However, it is not clear that there is a gcd-closed set S = {x1 , . . . , xn } with maxx∈S {|GS (x)|} = 1 such that the GCD matrix ((xi , xj )) on S does not divide the LCM matrix ([xi , xj ]) on S in the ring Mn (Z). We believe that the answer to this question should be negative. We give the following conjecture. Conjecture 4.4. Let S = {x1 , . . . , xn } be a gcd-closed set and f a completely multiplicative function such that (f ∗ µ)(d) is a nonzero integer whenever d | lcm(S). If maxx∈S {|GS (x)|} = 1, then the matrix (f (xi , xj )) on S divides the matrix (f [xi , xj ]) on S in the ring Mn (Z). The set S is said to be lcm-closed if [xi , xj ] ∈ S for all 1 i, j n. It is easy to see that a multiple-closed set is an lcm-closed set but not conversely. The set S is said to be odd-lcm-closed if S is lcm-closed and every element in S is an odd number. For x ∈ A, where A is a set of distinct positive integers, we say that m ∈ A is a least-type multiple of x in A, if x | m, m = x and the conditions x | y | m and y ∈ A imply that y ∈ {x, m}. We define LA (x) to be the set of all least-type multiples of x in A. Finally, we suggest the following similar conjectures as the conclusion of this paper. Conjecture 4.5. Let ε = 0 and let S = {x1 , . . . , xn } be an odd-lcm-closed set. Then the matrix ([xi , xj ]ε ) on S is nonsingular. Conjecture 4.6. Let S = {x1 , . . . , xn } be an odd-lcm-closed set and f a completely multiplicative function. If f is a strictly monotonous function, then the matrix (f (xi , xj )) on S is nonsingular. Conjecture 4.7. Let S = {x1 , . . . , xn } be an odd-lcm-closed set and f a completely multiplicative function. If f is a strictly monotonous function, then the matrix (f [xi , xj ]) on S is nonsingular. Conjecture 4.8. Let S = {x1 , . . . , xn } be a lcm-closed set and f a completely multiplicative function such that (f ∗ µ)(d) is a nonzero integer whenever d | lcm(S). If maxx∈S {|LS (x)|} = 1, then the matrix (f (xi , xj )) on S divides the matrix (f [xi , xj ]) on S in the ring Mn (Z).
Acknowledgments The author thanks Professor Keqin Feng for his valuable discussions and encouragement at one stage of this work during 2000. The author also thanks the anonymous referee for helpful comments and suggestions which led to a significant improvement of the presentation.
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References [1] T.M. Apostol, Arithmetical properties of generalized Ramanujan sums, Pacific J. Math. 41 (1972) 281–293. [2] T.M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976. [3] S. Beslin, S. Ligh, Another generalization of Smith’s determinant, Bull. Austral. Math. Soc. 40 (3) (1989) 413–415. [4] K. Bourque, S. Ligh, On GCD and LCM matrices, Linear Algebra Appl. 174 (1992) 65–74. [5] K. Bourque, S. Ligh, Matrices associated with arithmetical functions, Linear Multilinear Algebra 34 (1993) 261–267. [6] K. Bourque, S. Ligh, Matrices associated with classes of arithmetical functions, J. Number Theory 45 (1993) 367–376. [7] K. Bourque, S. Ligh, Matrices associated with classes of multiplicative functions, Linear Algebra Appl. 216 (1995) 267–275. [8] S. Hong, LCM matrix on an r-fold gcd-closed set, J. Sichuan Univ. Nat. Sci. Ed. 33 (6) (1996) 650–657. [9] S. Hong, On LCM matrices on gcd-closed sets, Southeast Asian Bull. Math. 22 (1998) 381–384. [10] S. Hong, Bounds for determinants of matrices associated with classes of arithmetical functions, Linear Algebra Appl. 281 (1998) 311–322. [11] S. Hong, On the Bourque–Ligh conjecture of least common multiple matrices, J. Algebra 218 (1999) 216– 228; A sketch was given in Adv. Math. (China) 25 (1996) 566–568. [12] S. Hong, Lower bounds for determinants of matrices associated with classes of arithmetical functions, Linear Multilinear Algebra 45 (1999) 349–358. [13] S. Hong, Gcd-closed sets and determinants of matrices associated with arithmetical functions, Acta Arith. 101 (2002) 321–332. [14] S. Hong, On the factorization of LCM matrices on gcd-closed sets, Linear Algebra Appl. 345 (2002) 225– 233. [15] S. Hong, Factorization of matrices associated with classes of arithmetical functions, Colloq. Math. 98 (2003) 113–123. [16] S. Hong, Notes on power LCM matrices, Acta Arith. 111 (2004) 165–177. [17] P.J. McCarthy, A generalization of Smith’s determinant, Canad. Math. Bull. 29 (1988) 109–113. [18] H.J.S. Smith, On the value of a certain arithmetical determinant, Proc. London Math. Soc. 7 (1875-1876) 208–212.
Nonsingularity of matrices associated with classes of arithmetical functions ✩ Shaofang Hong Mathematical College, Sichuan University, Chengdu 610064, PR China Department of Mathematics, Technion—IIT, Haifa 32000, Israel Received 10 April 2001
Communicated by Michel Broué
Abstract Let S = {x1 , . . . , xn } be a set of n distinct positive integers. Let f be an arithmetical function. Let [f (xi , xj )] denote the n × n matrix having f evaluated at the greatest common divisor (xi , xj ) of xi and xj as its i, j -entry and (f [xi , xj ]) denote the n × n matrix having f evaluated at the least common multiple [xi , xj ] of xi and xj as its i, j -entry. The set S is said to be gcd-closed if (xi , xj ) ∈ S for all 1 i, j n. For an integer x, let ν(x) denote the number of distinct prime factors of x. In this paper, by using the concept of greatest-type divisor introduced by S. Hong in [Adv. Math. (China) 25 (1996) 566–568; J. Algebra 218 (1999) 216–228], we obtain a new reduced formula for det f [(xi , xj )] if S is gcd-closed. Then we show that if S = {x1 , . . . , xn } is a gcd-closed set satisfying maxx∈S {ν(x)} 2, and if f is a strictly increasing (respectively decreasing) completely multiplicative function, or if f is a strictly decreasing (respectively increasing) completely multiplicative function satisfying 0 < f (p) 1/p (respectively f (p) p) for any prime p, then the matrix [f (xi , xj )] (respectively (f [xi , xj ])) defined on S is nonsingular. As a corollary, we show the following interesting result: The LCM matrix ([xi , xj ]) defined on a gcd-closed set is nonsingular if maxx∈S {ν(x)} 2. 2004 Elsevier Inc. All rights reserved. Keywords: Gcd-closed set; Greatest-type divisor; Arithmetical function; Nonsingularity
✩ Supported partially by an NNSF of China (Grant No. 10101015) and the Lady Davis Fellowship at the Technion. E-mail addresses: [email protected], [email protected].
0021-8693/$ – see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jalgebra.2004.07.026
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S. Hong / Journal of Algebra 281 (2004) 1–14
1. Introduction Let S = {x1 , . . . , xn } be a set of n distinct positive integers. The matrix having the greatest common divisor (xi , xj ) of xi and xj as its i, j -entry is called the greatest common divisor (GCD) matrix, denoted by [(xi , xj )]. The matrix having the least common multiple [xi , xj ] of xi and xj as its i, j -entry is called the least common multiple (LCM) matrix, denoted by ([xi , xj ]). The set is said to be factor-closed if it contains every divisor of x for any x ∈ S. H.J.S. Smith [18] showed that the determinant of the GCD matrix [(xi , xj )] on a factor-closed set S is the product ni=1 ϕ(xi ), where ϕ is Euler’s totient function. The set S is said to be gcd-closed if (xi , xj ) ∈ S for all 1 i, j n. It is clear that a factor-closed set is a gcd-closed set but not conversely. In [3], Beslin and Ligh generalized Smith’s result by showing that the determinant of GCD matrix [(xi , xj )] defined on a gcd-closed set S = {x1 , . . . , xn } is equal to the product nk=1 αk , where αk =
ϕ(d).
d|xk dxt , xt
In [4], Bourque and Ligh proved that the determinant of the LCM matrix ([xi , xj ]) defined on a gcd-closed set S = {x1 , . . . , xn } is equal to the product nk=1 xk2 βk , where βk =
g(d)
d|xk dxt , xt
with the arithmetical function g defined by g(m) = m1 d|m dµ(d), and the function µ is the Möbius function. Let f be an arithmetical function. Let [f (xi , xj )] denote the n×n matrix having f evaluated at the greatest common divisor (xi , xj ) of xi and xj as its i, j -entry and (f [xi , xj ]) denote the n × n matrix having f evaluated at the least common multiple [xi , xj ] of xi and xj as its i, j -entry. In [18], Smith also considered the determinant of the matrix [f (xi , xj )] on a factor-closed set S. It was shown to be the product nk=1 (f ∗ µ)(xk ), where f ∗ µ is the Dirichlet product of f and µ. Apostol [1] extended Smith’s result. McCarthy in [17] generalized Smith’s and Apostol’s results to the class of even functions (mod r). Then Bourque and Ligh [6] extended Smith’s and Apostol’s and McCarthy’s results to certain two-variable arithmetical function. Hong [12] improved the lower bounds for the determinants of matrices introduced by Bourque and Ligh. Hong [13] extended the results of Smith, of Apostol, of McCarthy, and of Bourque and Ligh. Bourque and Ligh [4] showed that the GCD matrix [(xi , xj )] on S divides the LCM matrix ([xi , xj ]) on S in the ring Mn (Z) of n × n matrices over the integers if S is factorclosed. Hong [14] proved that such a factorization is no longer true in general if S is gcdclosed. In fact, Hong showed that if n 3, then for any gcd-closed set S = {x1 , . . . , xn }, the GCD matrix on S divides the LCM matrix on S in the ring Mn (Z). For n 4, there exists a gcd-closed set S = {x1 , . . . , xn }, such that the GCD matrix on S does not divide the LCM matrix on S in the ring Mn (Z). From Bourque and Ligh’s result [7, Theorem 4],
S. Hong / Journal of Algebra 281 (2004) 1–14
3
we can see that if S is a factor-closed set and f is a multiplicative function such that (f ∗ µ)(d) ∈ Z∗ whenever d | lcm(S) where Z∗ := Z\{0} denotes the set of nonzero integers and lcm(S) means the least common multiple of all elements in S, then the matrix (f (xi , xj )) divides the matrix (f [xi , xj ]) in the ring Mn (Z). Recently, Hong [15] show that for any multiple-closed set S (namely, y ∈ S whenever x | y | lcm(S) for any x ∈ S), and for any divisor chain S (i.e. x1 | · · · | xn ), if f is a completely multiplicative function such that (f ∗ µ)(d) is a nonzero integer whenever d | lcm(S), then the matrix (f (xi , xj )) divides the matrix (f [xi , xj ]) in the ring Mn (Z). But such a factorization is no longer true if f is multiplicative. A general question, raised in a slightly different form by Bourque and Ligh at the end of [5] is to characterize the couples (S, f ) with S gcd-closed for which the matrix [f (xi , xj )] is nonsingular. They gave tool to attack this problem by giving the following formula, which is another generalization of Smith’s result. Theorem 1.1 [5]. The determinant of the matrix [f (xi , xj )] defined on a gcd-closed set S = {x1 , . . . , xn } is equal to x∈S αS,f (x), where
αS,f (x) :=
(f ∗ µ)(d) and
(1)
d|x d∈ / ES (x)
ES (x) := z ∈ Z+ : ∃y ∈ S, y < x, z | y with Z+ denoting the set of positive integers. Thus the problem is reduced to recognize if x∈S αS,f (x) = 0 or not. Let A be a set of positive integers. We denote by gcd(A) the greatest common divisor of the elements of A. Moreover, for x ∈ A, we say that d ∈ A is a greatest-type divisor of x in A, if d | x, d = x and the conditions d | y | x and y ∈ A imply that y ∈ {x, d} (see [11]). Note that the concept of greatest-type divisor introduced in [11] played a central role in our solution [11] to the Borque–Ligh conjecture [4]. We define GA (x) to be the set of all greatest-type divisors of x in A. Namely, we have GA (x) := d ∈ A: d | x, d = x and [d | y | x and y ∈ A] ⇒ y ∈ {x, d} . The aim of our paper is to continue the work of Bourque and Ligh [5] by giving the following formula. Theorem 1.2. For all gcd-closed sets S and all complex-valued functions f defined on S and x ∈ S, we have αS,f (x) =
(−1)|J | f gcd(J ∪ {x}) .
J ⊂GS (x)
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S. Hong / Journal of Algebra 281 (2004) 1–14
(Notice that if J = ∅, gcd(J ∪ {x}) = gcd(J ).) This formula makes the computation of αS,f (x) easy if |GS (x)| is small. For instance in the case where S is a divisor chain or when all the x are primes or one, because then |GS (x)| = 1. For an integer x, let ν(x) denote the number of distinct prime factors of x. If maxx∈S {ν(x)} 2, then GS (x) even if it is big, has a special structure that permits to give sufficient condition on f such that the matrix [f (xi , xj )] is nonsingular. An arithmetical function f (x) is said to be strictly increasing (respectively strictly decreasing) if we have f (x1 ) < f (x2 ) (respectively f (x1 ) > f (x2 )) whenever x1 < x2 and x1 , x2 ∈ Z+ . Theorem 1.3. Let S = {x1 , . . . , xn } be a gcd-closed set satisfying maxx∈S {ν(x)} 2 and f a completely multiplicative function. If f is a strictly increasing function or f is a strictly decreasing function satisfying 0 < f (p) 1/p for all primes p, then the matrix [f (xi , xj )] on S is nonsingular. As we shall see, it is easy to deduce the following result. Theorem 1.4. Let S = {x1 , . . . , xn } be a gcd-closed set satisfying maxx∈S {ν(x)} 2 and f a completely multiplicative function. If f is a strictly increasing function satisfying f (p) p for all primes p or f is a strictly decreasing function, then the matrix (f [xi , xj ]) on S is nonsingular. As a consequence, we have the following result. Theorem 1.5. Let S = {x1 , . . . , xn } be a gcd-closed set satisfying maxx∈S {ν(x)} 2 and let ε < 0 or ε 1. Then the matrix ([xi , xj ]ε ) on S is nonsingular. In particular, we have the following interesting result. Theorem 1.6. Let S = {x1 , . . . , xn } be a gcd-closed set satisfying maxx∈S {ν(x)} 2. Then the LCM matrix ([xi , xj ]) on S is nonsingular. Moreover, we will show by an example (see Section 4) that we cannot change 2 by 3 in this last theorem. Theorem 1.2 is proved in the second section and the proofs of Theorems 1.3–1.5 are given in the third section. The last section is devoted to some remarks and conjectures. Throughout this paper, we denote by |A| the cardinality of a finite set A.
2. Proof of Theorem 1.2 We [11] gave a reduction of the formula for the determinant of the LCM matrix ([xi , xj ]) by introducing the concept of the greatest-type divisor. In the present section we will prove Theorem 1.2 by using the similar ideas as in [8,11]. First one needs a generalization of the principle of cross-classification in [8] to give a preliminary reduction of the formula
S. Hong / Journal of Algebra 281 (2004) 1–14
5
for αS,f . Note that an alternative proof using induction was given in [14]. For the convenience of the reader, we here give another proof using the principal of cross-classification. In next lemma, for a subset T of R, we denote by T the set R \ T of the elements of R that are not in T . Moreover, j ∈φ Rj is meant to be R. Lemma 2.1 [8, Lemma 1]. Let R be any given finite set, {Ri }i∈I a finite family of subsets of R, and f a complex-valued function defined on R. Then f (x) = (−1)|J | f (x). (2) x∈
i∈I
J ⊂I
Ri
x∈
j∈J
Rj
Proof. First observe that for f = 1, (2) is the principal of cross-classification [2] that can be written as |J | Ri = (−1) Rj . J ⊂I
i∈I
j ∈J
Let A be a subset of R. Define the function 1A for x ∈ R by 1A (x) = 1 if x ∈ A, and 0 otherwise. By applying the principal of cross-classification to the family of |I | + 1 subsets, {Ri }i∈I and A, we get
|J | |J |+1 1A (x) = R i ∩ A = (−1) Rj + (−1) Rj ∩ A x∈
i∈I
Ri
J ⊂I
i∈I
j ∈J
J ⊂I
j ∈J
= (−1)|J | Rj − Rj ∩ A = (−1)|J | Rj ∩ A J ⊂I
=
J ⊂I
j ∈J
(−1)
j ∈J
|J | x∈
j∈J
J ⊂I
j ∈J
1A (x). Rj
This completes the proof of (2) for f = 1A . Now the formula f= f (y)1{x} y∈R
permits us to deduce (2) for general f , by linearity. 2 We can now give a preliminary reduction of the formula for αS,f (x). Claim 2.2. Let S = {x1 , . . . , xn } be gcd-closed and x ∈ S. Then (−1)|J | f gcd(J ∪ {x}) , αS,f (x) = J ⊂TS (x)
where TS (x) := {y: y ∈ S and y < x}.
(3)
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S. Hong / Journal of Algebra 281 (2004) 1–14
Proof. In Lemma 2.1, let R = {d ∈ Z+ : d | x}. For y ∈ TS (x), let RS (y) = {d ∈ R: d | y}. Then RS (y) = {d ∈ Z+ : d | (x, y)} and y∈TS (x) RS (y) = ES (x) ∩ R. By Lemma 2.1, we have (f ∗ µ)(d) = (−1)|J | (f ∗ µ)(d). (4) αS,f (x) = J ⊂TS (x)
d∈ES (x)∩R
Since d ∈
y∈J
d∈
y∈J
RS (y)
RS (y) implies that d | gcd(J ∪ {x}), we have by [10, Lemma 1]: d∈
y∈J
(f ∗ µ)(d) = f gcd(J ∪ {x}) .
(5)
RS (y)
It then follows from Eqs. (4) and (5) that (3) holds. This completes the proof of Claim 2.2. 2 Consequently, we give further reduction of the formula for αS,f (x). Claim 2.3. Let S = {x1 , . . . , xn } be gcd-closed and x ∈ S. Then
αS,f (x) =
(−1)|J | f gcd(J ∪ {x}) ,
(6)
J ⊂JS (x)
where JS (x) := TS (x) \ IS (x) and IS (x) := {y ∈ TS (x): y x}. Proof. If |IS (x)| = 0, then it follows from Claim 2.2 that Claim 2.3 holds. In what follows let |IS (x)| 1. Note that for y ∈ JS (x), we have y | x. Since S is gcd-closed, min(S) | x. Thus one has |JS (x)| 1. Note also that |IS (x)| + |JS (x)| = |TS (x)|. By Claim 2.2, we have αS,f (x) = f (x) + ∆ + ∆, where ∆ = (−1)|J | f gcd(J ∪ {x}) and
∆=
J ⊂JS (x) |J |1
(−1)|J1 |+|J2 | f gcd(J1 ∪ J2 ∪ {x}) .
(7)
(8)
J1 ⊂JS (x) J2 ⊂IS (x) |J1 |1 |J2 |1
For any given subset J2 ⊂ IS (x) with |J2 | 1, it follows from S is gcd-closed that gcd(J2 ∪ {x}) ∈ S. Let l = gcd(J2 ∪ {x}). Then l | x and l < x. So l ∈ JS (x). Thus we have by (8): ∆=
J2 ⊂IS (x) J1 ⊂JS (x) |J2 |1 |J1 |1
(−1)|J1 |+|J2 | f gcd(J1 ∪ J2 ∪ {x})
S. Hong / Journal of Algebra 281 (2004) 1–14
=
(−1)|J1 |+|J2 | f gcd(J1 ∪ J2 ∪ {x})
J2 ⊂IS (x) J1 ⊂JS (x) l ∈J / 1 |J2 |1
=
7
+ (−1)|J1 |+|J2 |+1 f gcd(J1 ∪ J2 ∪ {l} ∪ {x}) (−1)|J1 |+|J2 | f gcd(J1 ∪ {l}) + (−1)|J1 |+|J2 |+1 f gcd(J1 ∪ {l})
J2 ⊂IS (x) J1 ⊂JS (x) l ∈J / 1 |J2 |1
= 0.
(9)
Therefore, it follows from Eqs. (7) and (9) that (6) holds. The proof of Claim 2.3 is complete. 2 Now we can prove Theorem 1.2. Proof of Theorem 1.2. For the case |TS (x)| 1, Theorem 1.2 is clearly true. In what follows, let |TS (x)| 2. Let JS (x) = {y ∈ TS (x): y | x}. Then |JS (x)| 1. It is clear that GS (x) ⊂ JS (x). If |JS (x)| = 1, then JS (x) = {min(S)}. Note that |GS (x)| 1. So one has GS (x) = {min(S)} = JS (x). Thus by Claim 2.3, the result is true. In the following let |JS (x)| 2. Let LS (x) = JS (x)\GS (x). We assert that LS (x) = ∅. Otherwise, LS (x) = φ implies that GS (x) = JS (x). But min(S) ∈ JS (x). Then min(S) ∈ GS (x). From |JS (x)| 2, one deduces that there is y ∈ JS (x), such that y ∈ JS (x) = GS (x). Since S is gcdclosed, we have min(S) | y. This is absurd. Therefore, the assertion is true. In a similar way to (7), we have by Claim 2.3: αS,f (x) = f (x) + ∆ + ∆, where ∆ = (−1)|J | f gcd(J ∪ {x}) and
∆=
J ⊂GS (x) |J |1
J1 ⊂GS (x) J2 ⊂LS (x) |J2 |1
=
J2 ⊂LS (x) |J2 |1
(−1)|J2 |
(−1)|J1 |+|J2 | f gcd(J1 ∪ J2 ∪ {x})
(−1)|J1 | f gcd(J1 ∪ J2 ∪ {x}) .
(10)
J1 ⊂GS (x)
To prove Theorem 1.2, one needs only to show that ∆ = 0 which will be done in the following. For any given J2 ⊂ LS (x) with |J2 | 1, let P(x, J2) = {y ∈ GS (x): z | y for some z ∈ J2 } and let Q(x, J2 ) = GS (x) \ P(x, J2). Clearly we have 1 |P(x, J2 )| |GS (x)| and 0 |Q(x, J2 )| |GS (x)| − 1. Then
(−1)|J1 | f gcd(J1 ∪ J2 ∪ {x})
J1 ⊂GS (x)
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S. Hong / Journal of Algebra 281 (2004) 1–14
=
(−1)|J1 |+|J1 | f gcd J1 ∪ J1 ∪ J2 ∪ {x}
J1 ⊂Q(x,J2 ) J1 ⊂P (x,J2 )
=
(−1)|J1 |+|J1 | f gcd J1 ∪ J2 ∪ {x}
J1 ⊂Q(x,J2 ) J1 ⊂P (x,J2 )
(since by the definition of P(x, J2 ), we have gcd(J1 ∪ J2 ) = gcd(J2 ) for any J1 ⊂ P(x, J2 ))
=
(−1)|J1 | f gcd J1 ∪ J2 ∪ {x}
J1 ⊂Q(x,J2 )
=
=
J1 ⊂Q(x,J2 )
=
(−1)|J1 |
J1 ⊂P (x,J2 )
(−1)
|J1 |
|P (x,J2 )| f gcd J1 ∪ J2 ∪ {x} 1 + (−1)r
J1 ⊂Q(x,J2 )
r=1
(−1)
|J1 |
1
J1 ⊂P (x,J2 ) |J1 |=r
|P (x,J2 )| r |P(x, J2 )| f gcd J1 ∪ J2 ∪ {x} 1 + (−1) r r=1
(−1)|J1 | f gcd J1 ∪ J2 ∪ {x} · (1 − 1)|P (x,J2)|
J1 ⊂Q(x,J2 )
= 0.
(11)
It then follows from Eqs. (10) and (11) that ∆ = 0. This completes the proof of Theorem 1.2. 2
3. Proofs of Theorems 1.3–1.5 Throughout this section, let S = {x1 , . . . , xn } be a gcd-closed set. Let αS,f (x) be defined as in (1). It is clear that αS,f (min(S)) = f (min(S)). We have the following lemmas. Lemma 3.1. Let x = pe q h ∈ S, where p and q are distinct primes, e and h are positive integers. Then GS (x) must have the form {pe1 q h1 , . . . , pem q hm }, where 1 m min{e, h}, 0 e1 < · · · < em e, h h1 > · · · > hm 0, and ei + hi e + h − 1 (i = 1, . . . , m). Proof. Let |GS (x)| = m. Since x = pe q h , one may let GS (x) = {pe1 q h1 , . . . , pem q hm }, where ei and hi (1 i m) are nonnegative integers satisfying 0 ei e, 0 hi h and ei + hi e + h − 1. We claim that for any i, j ∈ {1, . . . , m}, i = j , we have ei = ej . Otherwise, suppose that there exist i, j ∈ {1, . . . , m}, i = j , such that ei = ej . Then pei q hi | pej q hj or pej q hj | pei q hi . This contradicts to the fact that pei q hi and pej q hj are the greatest-type divisors of x in S. Thus ei = ej for any i, j ∈ {1, . . . , m}, i = j . Similarly, for i, j ∈ {1, . . . , m}, i = j , we have hi = hj .
S. Hong / Journal of Algebra 281 (2004) 1–14
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Without loss of generality, one may assume that 0 e1 < e2 < · · · < em . Since pe1 q h1 , . . . , pem q hm are the greatest-type divisors, then for any i, j ∈ {1, . . . , m}, i = j , we have both pei q hi pej q hj and pej q hj pei q hi . Therefore for any i ∈ {1, 2, . . . , m − 1}, it follows from ei < ei+1 and pei q hi pei+1 q hi+1 that hi > hi+1 . So h1 > h2 > · · · > hm 0. It is clear that for 1 i m, we have ei + hi e + h. Suppose that there exists 1 i m such that ei + hi = e + h. One can deduce that ei = e and hi = h. So we have pei q hi = x. This contradicts that pei q hi is a greatest-type divisor of x. Then for i = 1, 2, . . . , m, we have ei + hi e + h − 1. The proof of Lemma 3.1 is complete. 2 Lemma 3.2. Let x = pe q h ∈ S, where p and q are distinct primes, e and h are positive integers. If GS (x) = {pe1 q h1 , . . . , pem q hm }, where 1 m min{e, h}, 0 e1 < · · · < em e, h h1 > · · · > hm 0, and ei + hi e + h − 1 (i = 1, . . . , m), then we have e h e h 1 1 f p q − f p q , m αS,f (x) = e h m−1 e h f p iq i + f pei q hi+1 , f p q − i=1
if m = 1, if m 2.
i=1
Proof. By Lemma 3.1, we know that GS (x) = pej q hj : 1 j m with 0 e1 < · · · < em e and h h1 > · · · > hm 0. We can write the parts J of GS (x) as J = pei q hi : i ∈ I
with I ⊂ {1, . . . , m}.
With the notation (emin(φ) , hmin(φ) ) = (e, h), Theorem 1.2 gives
αS,f (x) =
(−1)|I | f pemin(I ) q hmax(I )
I ⊂{1,...,m}
= f (x) +
f pea q hb
(−1)|I | .
min(I )=a max(I )=b
1abm
We conclude by the observation that for b a + 2, we have min(I )=a max(I )=b
(−1)|I | =
I ⊂{a+1,...,b−1}
This completes the proof of Lemma 3.2. 2 We are now in a position to prove Theorem 1.3.
(−1)|I | = 0.
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Proof of Theorem 1.3. Without loss of any generality, one may let x1 < · · · < xn . First we note that f (x) > 0 for any integer x > 0, since f is completely multiplicative satisfying f (p) > 0 for all prime p. Since maxx∈S {ν(x)} 2, then for x ∈ S with x > 1, we have ν(x) = 1 or 2. That is, we have x = pe , where e 1 is an integer and p is a prime, or x = pe q h , where e 1 and h 1 are integers, p and q are distinct primes. We claim that for x ∈ S, αS,f (x) = 0. If x = x1 , then by Theorem 1.2 we have αS,f (x) = f (x1 ) = 0. The claim is true. In the following, let x = x1 . Consider the following two cases. Case 1. x = pe . Then x has only one greatest-type divisor in S whose form must be pl , where l is an integer and 0 l e − 1. By Theorem 1.2, we have αS,f (x) = f (pe ) − f (pl ). Since f is strictly increasing or strictly decreasing, one then deduce that αS,f (x) > 0 or αS,f (x) < 0, respectively. The claim is true. Case 2. x = pe q h . It follows from Lemmas 3.1 and 3.2 that there exist 2m, where 1 m min{e, h}, integers e1 , . . . , em , h1 , . . . , hm satisfying 0 e1 < · · · < em e, h h1 > · · · > hm 0, and ei + hi e + h − 1 (i = 1, . . . , m), such that GS (x) = {pe1 q h1 , . . . , pem q hm }. If m = 1, then by Lemma 3.2 we have αS,f (x) = f (pe q h ) − f (pe1 q h1 ). By the assumption that f is strictly decreasing or strictly increasing, we have αS,f (x) < 0 or αS,f > 0, respectively. The claim is true. In the following let m 2. Let f be a strictly increasing function. Then for primes p and q, f (p) > f (1) = 1 and f (q) > 1. Since em e, h1 h, we have f (pe q h ) f (pem q h1 ). Therefore, m m−1 αS,f (x) f pem q h1 − f pei q hi + f pei q hi+1 i=1
=
m−1
i=1
e h f p m q i − f pem q hi+1 − f pei q hi + f pei q hi+1
i=1
=
m−1
f pei q hi+1 f (p)em −ei − 1 f (q)hi −hi+1 − 1 > 0.
i=1
So the claim for this case is true. Now let f be a strictly decreasing function satisfying 0 < f (p) 1/p for all primes p. If m = 2, then by Lemma 3.2 we have αS,f (x) = f pe q h + f pe1 q h2 − f pe1 q h1 − f pe2 q h2 . Since f is completely multiplicative, we have αS,f (x) = f pe q h + f pe2 q h1 f pe1 −e2 q h2 −h1 − f pe1 −e2 − f q h2 −h1 1 1 1 − − . = f pe q h + f pe2 q h1 f (pe2 −e1 q h1 −h2 ) f (pe2 −e1 ) f (q h1 −h2 )
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Since f satisfies 0 < f (p) 1/p for all primes p, one can then deduce that 1 f (pe2 −e1 q h1 −h2 )
−
1 f (pe2 −e1 )
−
1 f (q h1 −h2 )
=
1
−1
1
f (pe2 −e1 ) f (q h1 −h2 ) pe2 −e1 − 1 q h1 −h2 − 1 − 1
−1 −1
(2 − 1)(3 − 1) − 1 = 1 > 0. Thus αS,f (x) > 0. The claim holds for this case. In what follows let m 3. Then we have by Lemma 3.2: αS,f (x) = f pe q h + f pe1 q h2 − f pe1 q h1 − f pe2 q h2 +
m e f p i−1 q hi − f pei q hi . i=3
By the above arguments, one knows that f pe1 q h2 − f pe1 q h1 − f pe2 q h2 > 0. Since f is strictly decreasing, we have for 3 i m: f pei−1 q hi − f pei q hi > 0. So αS,f (x) > 0. The claim is true in this case. It then follows from the claim and Theorem 1.1 that det[f (xi , xj )] = 0. Therefore the matrix [f (xi , xj )] is nonsingular. The proof of Theorem 1.3 is complete. 2 Finally we give the proofs of Theorems 1.4 and 1.5. Proof of Theorem 1.4. Define for any integer x the function 1/f to be 0 if f (x) = 0; 1/f (x) if f (x) = 0. Clearly we have
1 (f [xi , xj ]) = diag{f (x1 ), . . . , f (xn )} · (xi , xj ) · diag{f (x1 ), . . . , f (xn )}. f Then Theorem 1.3 applied to the function 1/f gives the desired result.
2
Proof of Theorem 1.5. Let f (x) = x ε in Theorem 1.4, where ε < 0 or ε 1. Then the result follows immediately from Theorem 1.4. 2
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4. Remarks and conjectures In [4], Bourque and Ligh conjectured that the LCM matrix ([xi , xj ]) defined on any gcd-closed set S = {x1 , . . . , xn } is nonsingular. In [8,11], by introducing the concept of greatest-type divisor to reduce greatly the formula for det[xi , xj ], we showed that this conjecture is true for n 7 and is not true for n 8. We showed also, in [9], that this conjecture is true for certain gcd-closed sets. From Theorem 1.6, one then knows that the Bourque–Ligh conjecture is true for the gcd-closed sets S = {x1 , . . . , xn } so that each of S has at most two distinct prime factors. However, if S is a gcd-closed set satisfying maxx∈S {ν(x)} 3, then the LCM matrix [(xi , xj )] need not be nonsingular. For example, let S = {1, 2, 5, 8, 10, 13, 26, 65, 520}.
(12)
Then S is gcd-closed and maxx∈S {ν(x)} = 3. Since GS (520) = {8, 10, 26, 65}, we have β9 (1, 2, 5, 8, 10, 13, 26, 65, 520) =
1 1 1 1 1 1 1 − − − − + + = 0. 520 8 10 26 65 5 13
Thus the LCM matrix ([xi , xj ]) on S is singular. We believe that Theorems 1.3 and 1.4 still hold if the conditions “satisfying 0 < f (p) 1/p for all primes p” and “satisfying f (p) p for all primes p” are suppressed, respectively. We believe also that the condition “ε < 0 or ε 1” in Theorem 1.5 could be improved to “ε = 0.” Unfortunately, so far we have not found the proof yet. The set S is said to be odd-gcd-closed if S is gcd-closed and every element in S is an odd number. The set S is said to be even-gcd-closed if S is not odd-gcd-closed. By [11] and (12), we know that there are even-gcd-closed sets S such that the LCM matrix ([xi , xj ]) on S is singular. But it is not clear that there is an odd-gcd-closed set S such that the LCM matrix ([xi , xj ]) on S is singular. We believe that the answer to this question should be negative. We raise the following conjecture which generalizes Conjecture 5.1 of [16]. Conjecture 4.1. Let ε = 0 and let S = {x1 , . . . , xn } be an odd-gcd-closed set. Then the matrix ([xi , xj ]ε ) on S is nonsingular. An arithmetical function f is called strictly monotonous if it is strictly increasing or strictly decreasing. Furthermore, we propose the following conjectures. Conjecture 4.2. Let S = {x1 , . . . , xn } be an odd-gcd-closed set and f a completely multiplicative function. If f is a strictly monotonous function, then the matrix (f (xi , xj )) on S is nonsingular. Conjecture 4.3. Let S = {x1 , . . . , xn } be an odd-gcd-closed set and f a completely multiplicative function. If f is a strictly monotonous function, then the matrix (f [xi , xj ]) on S is nonsingular.
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According to [14], there is a gcd-closed set S = {x1 , . . . , xn } with maxx∈S {|GS (x)|} = 2 such that the GCD matrix ((xi , xj )) on S does not divide the LCM matrix ([xi , xj ]) on S in the ring Mn (Z). However, it is not clear that there is a gcd-closed set S = {x1 , . . . , xn } with maxx∈S {|GS (x)|} = 1 such that the GCD matrix ((xi , xj )) on S does not divide the LCM matrix ([xi , xj ]) on S in the ring Mn (Z). We believe that the answer to this question should be negative. We give the following conjecture. Conjecture 4.4. Let S = {x1 , . . . , xn } be a gcd-closed set and f a completely multiplicative function such that (f ∗ µ)(d) is a nonzero integer whenever d | lcm(S). If maxx∈S {|GS (x)|} = 1, then the matrix (f (xi , xj )) on S divides the matrix (f [xi , xj ]) on S in the ring Mn (Z). The set S is said to be lcm-closed if [xi , xj ] ∈ S for all 1 i, j n. It is easy to see that a multiple-closed set is an lcm-closed set but not conversely. The set S is said to be odd-lcm-closed if S is lcm-closed and every element in S is an odd number. For x ∈ A, where A is a set of distinct positive integers, we say that m ∈ A is a least-type multiple of x in A, if x | m, m = x and the conditions x | y | m and y ∈ A imply that y ∈ {x, m}. We define LA (x) to be the set of all least-type multiples of x in A. Finally, we suggest the following similar conjectures as the conclusion of this paper. Conjecture 4.5. Let ε = 0 and let S = {x1 , . . . , xn } be an odd-lcm-closed set. Then the matrix ([xi , xj ]ε ) on S is nonsingular. Conjecture 4.6. Let S = {x1 , . . . , xn } be an odd-lcm-closed set and f a completely multiplicative function. If f is a strictly monotonous function, then the matrix (f (xi , xj )) on S is nonsingular. Conjecture 4.7. Let S = {x1 , . . . , xn } be an odd-lcm-closed set and f a completely multiplicative function. If f is a strictly monotonous function, then the matrix (f [xi , xj ]) on S is nonsingular. Conjecture 4.8. Let S = {x1 , . . . , xn } be a lcm-closed set and f a completely multiplicative function such that (f ∗ µ)(d) is a nonzero integer whenever d | lcm(S). If maxx∈S {|LS (x)|} = 1, then the matrix (f (xi , xj )) on S divides the matrix (f [xi , xj ]) on S in the ring Mn (Z).
Acknowledgments The author thanks Professor Keqin Feng for his valuable discussions and encouragement at one stage of this work during 2000. The author also thanks the anonymous referee for helpful comments and suggestions which led to a significant improvement of the presentation.
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